Quantile-Based Fuzzy Semi-Parametric Support Vector Regression Model Based on Autoregressive Fuzzy Terms: An Estimation Approach
Batoul Salamah
1
(
Department of Applied Mathematics, PhD candidate in Applied Mathematics, Rasht, Iran.
)
Reza Zarei
2
(
Department of Statistics Assistant, Professor of Statistics, Rasht, Iran.
)
Farshid Mehrdoust
3
(
Department of Applied Mathematics, Professor of Applied Mathematics, Rasht, Iran.
)
Mohammad Ghasem Akbari
4
(
Department of Statistics, Associate Professor of Statistics, Birjand, Iran.
)
الکلمات المفتاحية: Fuzzy semi-parametric model, Support vector machine, Quantile regression, Fuzzy correlated errors, Goodness-of-fit measure, Kernel function.,
ملخص المقالة :
The semi-parametric regression model is one of the most useful statistical tools that has gained significant attention recently due to its key capability of combining both parametric and nonparametric features in one model. In practical applications, however, the recorded information or the relationship between one or more independent variables and the dependent variable is typically imprecise. Additionally, in certain situations, the error terms exhibit heteroscedasticity or the data distribution is skewed, which leads to inaccurate results when using conventional least squares models. In this regard, this paper introduces a semi-parametric quantile-based regression model using the support vector machine technique, along with precise regressors and fuzzy outcomes. We also employ the classic Durbin-Watson test to explore the existence of correlation among fuzzy residual expressions. We propose a mixed procedure that incorporates a mean absolute error and a cross-validation measure to calculate fuzzy multipliers in addition to the unknown autocorrelation criteria. We illustrate the efficacy of the suggested method via three numerical examples, including two applied examples and a simulation study. To this end, we use some common goodness-of-fit measurements to assess the performance of the suggested method in comparison with other approaches. The numerical results showed that when the fuzzy error terms are correlated, the suggested fuzzy semi-parametric quantile-based regression model performs better than the other methods.
[1] Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning. Springer; 2009.
[2] Vapnik VN. The Nature of Statistical Learning Theory. Springer; 1995.
[3] Vapnik VN. Statistical Learning Theory. Wiley; 1998.
[4] Gu B, Sheng VS, Wang Z, Ho D, Osman S, Li S. Incremental learning for v-support vector regression. Neural Networks. 2015; 67: 140-150. DOI: https://doi.org/10.1016/j.neunet.2015.03.013
[5] Gu B, Sheng VS, Tay KY, Romano W, Li S. Cross validation through two-dimensional solution surface for cost-sensitive SV M. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2016; 39(6): 1103-1121. DOI: https://doi.org/10.1109/TPAMI.2016.2578326
[6] Koenker R, Bassett JRG. Regression quantiles. Econometrica: Journal of the Econometric Society. 1978; 46(1): 33-50. DOI: https://doi.org/10.2307/1913643
[7] Koenker R. Quantile Regression. Cambridge University Press; 2005.
[8] Coppi R. Management of uncertainty in statistical reasoning: The case of regression analysis. International Journal of Approximate Reasoning. 2008; 47(3), 284-305. DOI: https://doi.org/10.1016/j.ijar.2007.05.011
[9] Zadeh LA. Fuzzy sets. Information and Control. 1965; 8(3): 338-353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X
[10] Taheri SM. Trends in fuzzy statistics. Austrian Journal of Statistics. 2003; 32(3): 239-257. DOI: https://doi.org/10.17713/ajs.v32i3.459
[11] Zendehdel J, Rezaei M, Akbari MG, Zarei R, Alizadeh Noughabi H. Testing exponentiality for imprecise data and its application. Soft Computing. 2018; 22: 3301-3312. DOI: https://doi.org/10.1007/s00500- 017-2566-y
[12] Tanaka H, Uejima S, Asai K. Linear regression analysis with fuzzy model. IEEE Transaction on Systems, Man, and Cybernetics. 1982; 12(6): 903-907. DOI: https://doi.org/10.1109/TSMC.1982.4308925
[13] Kacprzyk J, Fedrizzi M. Fuzzy Regression Analysis. Warsaw : Omnitech Press; 1992.
[14] Naderkhani R, Behzad MH, Razzaghnia T, Farnoosh R. Fuzzy regression analysis based on fuzzy neural
networks using trapezoidal data. International Journal of Fuzzy Systems. 2021; 23(5): 1267-1280. DOI: https://doi.org/10.1007/s40815-020-01033-2
[15] Chachi J, Taheri SM, DUrso P. Fuzzy regression analysis based on M-estimates. Expert Systems with Applications. 2022; 187: 115891. DOI: https://doi.org/10.1016/j.eswa.2021.115891
[16] Yoon JH, Kim DJ, Koo YY. Novel fuzzy correlation coefficient and variable selection method for fuzzy regression analysis based on distance approach. International Journal of Fuzzy Systems. 2023; 25(8): 2969-2985. DOI: https://doi.org/10.1007/s40815-023-01546-6
[17] Baha Alwan, E., Abdulmohsin Ali, O. About semi-parametric methodology for fuzzy quantile regression model estimation: A review. Journal of Economics and Administrative Sciences. 2024; 29(138): 127-146.
[18] Hesamian G, Akbari MG. Fuzzy quantile linear regression model adopted with a semi-parametric technique based on fuzzy predictors and fuzzy responses. Expert Systems with Applications. 2019; 118: 585-597. DOI: https://doi.org/10.1016/j.eswa.2018.10.026
[19] Arefi M. Quantile fuzzy regression based on fuzzy outputs and fuzzy parameters. Soft Computing. 2020; 24(1): 311-320. DOI: https://doi.org/10.1007/s00500-019-04424-2
[20] Chachi J, Chaji A. An OWA-based approach to quantile fuzzy regression. Computers & Industrial Engineering. 2021; 159: 107498. DOI: https://doi.org/10.1016/j.cie.2021.107498
[21] Khammar AH, Arefi M, Akbari MG. Quantile fuzzy varying coefficient regression based on kernel function. Applied Soft Computing. 2021; 107: 107313. DOI: https://doi.org/10.1016/j.asoc.2021.107313
[22] Hesamian G, Akbari MG. A fuzzy empirical quantile-based regression model based on triangular fuzzy numbers. Computational and Applied Mathematics. 2022; 41(6): 267. DOI: https://doi.org/10.1007/s40314-022-01974-4
[23] Hesamian G, Akbari MG, Asadollahi M. Fuzzy semi-parametric partially linear model with fuzzy inputs and fuzzy outputs. Expert Systems with Applications. 2017; 71: 230-239. DOI: https://doi.org/10.1016/j.eswa.2016.11.032
[24] Akbari MG, Hesamian G. Elastic net oriented to fuzzy semiparametric regression model with fuzzy explanatory variables and fuzzy responses. IEEE Transactions on Fuzzy Systems. 2019; 27(12): 2433- 2442. DOI: https://doi.org/10.1109/TFUZZ.2019.2900603
[25] Zadeh LA. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems. 1978; 1(1): 3-28. DOI:
https://doi.org/10.1016/0165-0114(78)90029-5
[26] Dubois D, Prade H. Operations on fuzzy numbers. International Journal of Systems Science. 1978; 9(6): 613-626. DOI: https://doi.org/10.1080/00207727808941724
[27] Lai YJ, Hwang CL. Fuzzy mathematical programming. In: Fuzzy Mathematical Programming. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, Heidelberg; 1992. p.74-186. DOI: https://doi.org/10.1007/978-3-642-48753-8 3
[28] Hesamian G, Akbari MG. Fuzzy absolute error distance measure based on a generalised difference operation. International Journal of Systems Science. 2018; 49(11): 2454-2462. DOI: https://doi.org/10.1080/00207721.2018.1505002
[29] Puri ML, Ralescu DA. Differentials of fuzzy functions. Journal of Mathematical Analysis and Applications. 1983; 91(2): 552-558. DOI: https://doi.org/10.1016/0022-247X(83)90169-5
[30] Gyrfi L, Kohler M, Krzyzak A, Walk H. A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics; 2006.
[31] Wasserman L. All of Nonparametric Statistics. Springer New York; 2006.
[32] Kutner MH, Nachtsheim CJ, Neter J. Applied Linear Regression Models. New York: McGraw-Hill Irwin; 2004.
[33] Durbin J, Watson GS. Testing for serial correlation in least squares regression. III. Biometrika. 1971; 58(1): 1-19. DOI: https://doi.org/10.1093/biomet/58.1.1
[34] Chatterjee S, Simonoff JS. Handbook of Regression Analysis. John Wiley & Sons; 2013.
[35] Akbari MG, Hesamian G. A fuzzy linear regression model with autoregressive fuzzy errors based on exact predictors and fuzzy responses. Computational and Applied Mathematics. 2022; 41(6): 284. DOI:https://doi.org/10.1007/s40314-022-01994-0
[36] Choi SH, Yoon JH. General fuzzy regression using least squares method. International Journal of Systems Science. 2010; 41(5): 477-485. DOI: https://doi.org/10.1080/00207720902774813
[37] Choi SH, Buckley JJ. Fuzzy regression using least absolute deviation estimators. Soft computing. 2008; 12: 257-263. DOI: https://doi.org/10.1007/s00500-007-0198-3
[38] Zeng W, Feng Q, Li J. Fuzzy least absolute linear regression. Applied Soft Computing. 2017; 52: 1009- 1019. DOI: https://doi.org/10.1016/j.asoc.2016.09.029
[39] Taheri SM, Kelkinnama M. Fuzzy linear regression based on least absolutes deviations. Iranian Journal of Fuzzy Systems. 2012; 9(1):121-140. DOI: https://doi.org/10.22111/ijfs.2012.230
[40] Kula KS, Apaydin A. Fuzzy robust regression analysis based on the ranking of fuzzy sets. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 2008; 16(05): 663-681. DOI: https://doi.org/10.1142/S0218488508005558
[41] Icen D, Demirhan H. Error measures for fuzzy linear regression: Monte Carlo simulation approach. Applied Soft Computing. 2016; 46: 104-114. DOI: https://doi.org/10.1016/j.asoc.2016.04.013
[42] Zhou J, Zhang H, Gu Y, Pantelous AA. Affordable levels of house prices using fuzzy linear regression analysis: The case of Shanghai. Soft Computing. 2018; 22: 5407-5418. DOI: https://doi.org/10.1007/s00500- 018-3090-4
[43] Hao PY, Chiang JH. Fuzzy regression analysis by support vector learning approach. IEEE Transactions on Fuzzy Systems. 2008; 16(2): 428-441. DOI: https://doi.org/10.1109/TFUZZ.2007.896359
